[toxik]

I think the usual abstract algebra definition is one factor: itself. 1 is not considered a factor so that prime factorizations are unique, otherwise you could tack on an infinity of ones.

Also Möbius becomes strange with infty factors.

[ginnungagap]

The abstract algebra definition is that p is a prime if whenever p divides a product ab then p divides at least one of a and b.

Having no proper factors is the definition of an irreducible element.

The two definitions agree for the integers and nice algebraic structure (UFDs) but there are algebraic structures in which they are not the same which explains why there are two differenr definitions and names in abstract algebra